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Properties

Label 546.4.c.c
Level $546$
Weight $4$
Character orbit 546.c
Analytic conductor $32.215$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,4,Mod(337,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 944x^{10} + 279558x^{8} + 29817396x^{6} + 1258307809x^{4} + 20904528132x^{2} + 108456931584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{6} q^{2} - 3 q^{3} - 4 q^{4} + (\beta_{6} + \beta_1) q^{5} - 6 \beta_{6} q^{6} + 7 \beta_{6} q^{7} - 8 \beta_{6} q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{6} q^{2} - 3 q^{3} - 4 q^{4} + (\beta_{6} + \beta_1) q^{5} - 6 \beta_{6} q^{6} + 7 \beta_{6} q^{7} - 8 \beta_{6} q^{8} + 9 q^{9} + (2 \beta_{2} - 2) q^{10} + (\beta_{11} + \beta_{8} + 5 \beta_{6}) q^{11} + 12 q^{12} + (\beta_{10} - 6 \beta_{6} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + (9 \beta_{11} + 9 \beta_{8} + 45 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{3} - 48 q^{4} + 108 q^{9} - 24 q^{10} + 144 q^{12} + 12 q^{13} - 168 q^{14} + 192 q^{16} + 130 q^{17} - 124 q^{22} + 8 q^{23} - 400 q^{25} + 144 q^{26} - 324 q^{27} + 12 q^{29} + 72 q^{30} - 84 q^{35} - 432 q^{36} + 216 q^{38} - 36 q^{39} + 96 q^{40} + 504 q^{42} - 240 q^{43} - 576 q^{48} - 588 q^{49} - 390 q^{51} - 48 q^{52} - 598 q^{53} + 814 q^{55} + 672 q^{56} - 1450 q^{61} + 364 q^{62} - 768 q^{64} - 682 q^{65} + 372 q^{66} - 520 q^{68} - 24 q^{69} - 1404 q^{74} + 1200 q^{75} - 434 q^{77} - 432 q^{78} + 3394 q^{79} + 972 q^{81} + 1240 q^{82} - 36 q^{87} + 496 q^{88} - 216 q^{90} + 504 q^{91} - 32 q^{92} - 2100 q^{94} + 1392 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 944x^{10} + 279558x^{8} + 29817396x^{6} + 1258307809x^{4} + 20904528132x^{2} + 108456931584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 919681 \nu^{10} - 772757405 \nu^{8} - 176926772703 \nu^{6} - 8431277925027 \nu^{4} + \cdots - 29\!\cdots\!00 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3989894737849 \nu^{10} + \cdots - 27\!\cdots\!12 ) / 56\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4291341310753 \nu^{10} + \cdots + 21\!\cdots\!00 ) / 18\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13889873578205 \nu^{10} + \cdots - 92\!\cdots\!92 ) / 28\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2257873775 \nu^{11} + 2055713667508 \nu^{9} + 567584014122990 \nu^{7} + \cdots + 48\!\cdots\!96 \nu ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 92\!\cdots\!83 \nu^{11} + \cdots - 12\!\cdots\!92 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!75 \nu^{11} + \cdots - 17\!\cdots\!12 \nu ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 68\!\cdots\!43 \nu^{11} + \cdots + 12\!\cdots\!92 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!43 \nu^{11} + \cdots - 12\!\cdots\!92 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 51\!\cdots\!45 \nu^{11} + \cdots + 38\!\cdots\!56 \nu ) / 38\!\cdots\!04 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + 3\beta_{3} - 7\beta_{2} - 158 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -15\beta_{11} - 40\beta_{9} - 18\beta_{8} - 40\beta_{7} + 962\beta_{6} - 362\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -319\beta_{10} + 319\beta_{9} + 486\beta_{5} - 11\beta_{4} - 1346\beta_{3} + 4290\beta_{2} + 55642 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8740 \beta_{11} + 1360 \beta_{10} + 19414 \beta_{9} + 15886 \beta_{8} + 18054 \beta_{7} + \cdots + 154181 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 122181 \beta_{10} - 122181 \beta_{9} - 229775 \beta_{5} - 99915 \beta_{4} + 555905 \beta_{3} + \cdots - 23293666 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4432621 \beta_{11} - 1054760 \beta_{10} - 8618672 \beta_{9} - 9704994 \beta_{8} + \cdots - 69003216 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 50893131 \beta_{10} + 50893131 \beta_{9} + 108106680 \beta_{5} + 74468957 \beta_{4} + \cdots + 10305598618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2183148290 \beta_{11} + 630101920 \beta_{10} + 3868422358 \beta_{9} + 5266235958 \beta_{8} + \cdots + 31672454819 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22200537113 \beta_{10} - 22200537113 \beta_{9} - 51106166225 \beta_{5} - 43231484315 \beta_{4} + \cdots - 4694268049650 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1063361781523 \beta_{11} - 340317509936 \beta_{10} - 1771090473712 \beta_{9} + \cdots - 14770502221742 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 17.3757i
− 6.33306i
− 4.64643i
− 3.08480i
9.52973i
21.9102i
− 21.9102i
− 9.52973i
3.08480i
4.64643i
6.33306i
17.3757i
− 2.00000i −3.00000 −4.00000 − 18.3757i 6.00000i − 7.00000i 8.00000i 9.00000 −36.7513
337.2 − 2.00000i −3.00000 −4.00000 − 7.33306i 6.00000i − 7.00000i 8.00000i 9.00000 −14.6661
337.3 − 2.00000i −3.00000 −4.00000 − 5.64643i 6.00000i − 7.00000i 8.00000i 9.00000 −11.2929
337.4 − 2.00000i −3.00000 −4.00000 − 4.08480i 6.00000i − 7.00000i 8.00000i 9.00000 −8.16960
337.5 − 2.00000i −3.00000 −4.00000 8.52973i 6.00000i − 7.00000i 8.00000i 9.00000 17.0595
337.6 − 2.00000i −3.00000 −4.00000 20.9102i 6.00000i − 7.00000i 8.00000i 9.00000 41.8204
337.7 2.00000i −3.00000 −4.00000 − 20.9102i − 6.00000i 7.00000i − 8.00000i 9.00000 41.8204
337.8 2.00000i −3.00000 −4.00000 − 8.52973i − 6.00000i 7.00000i − 8.00000i 9.00000 17.0595
337.9 2.00000i −3.00000 −4.00000 4.08480i − 6.00000i 7.00000i − 8.00000i 9.00000 −8.16960
337.10 2.00000i −3.00000 −4.00000 5.64643i − 6.00000i 7.00000i − 8.00000i 9.00000 −11.2929
337.11 2.00000i −3.00000 −4.00000 7.33306i − 6.00000i 7.00000i − 8.00000i 9.00000 −14.6661
337.12 2.00000i −3.00000 −4.00000 18.3757i − 6.00000i 7.00000i − 8.00000i 9.00000 −36.7513
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.c.c ✓ 12
13.b even 2 1 inner 546.4.c.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.c.c ✓ 12 1.a even 1 1 trivial
546.4.c.c ✓ 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 950T_{5}^{10} + 293913T_{5}^{8} + 34314644T_{5}^{6} + 1764934144T_{5}^{4} + 39604332304T_{5}^{2} + 307279531584 \) acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 307279531584 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( (T^{6} - 65 T^{5} + \cdots - 16945334784)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots - 683656030512)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 2182104943200)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 12371198574592)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 149544463465728)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 171623707406208)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 83\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
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